Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Abstract

Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

Transition from the Wave Equation to Either the Heat or the Transport Equations through Fractional Differential Expressions

“…However, such a case is out of the scope of the present work and will be studied elsewhere. Preliminary results associated to (11) have been already reported by other authors in e.g. [12][13][14][15][16][17] (see also the general approaches discussed in [7,11,[18][19][20][21]).…”

“…Additionally, the perturbations take negative values producing the presence of nodes as they propagate. The latter is markedly notorious in figure (i), where (α, β) is closest to vertex C. As we Figure 3: Propagation of the (right hand side) maxima of the solutions to the time-fractional differential equation (11) with initial conditions (54) and ϕ 2 = 0. We have taken u e (x, t) defined in (55)-(56) with β = 2, see Figure 2.…”

“…The constant k β > 0, measured in units [k β ] = [L] β [T ] −1 , is such that k 2 = k. Equation (12) is useful to describe symmetric Lévy-Feller diffusion processes in which jump components are present [11,22] (see also [23,24], and general approaches in [7,11,[18][19][20][21]). The parameter β = 2 gives the heat equation and β = 1 leads to a modified version of the one-dimensional transport equation [25,26]…”

Concept of dynamic memory in economics

“…The use of the concept of dynamic memory, which is described in this paper, allows us to build mathematical models of economic processes with different types of memory. The use of differential equations with derivatives of non-integer orders can allow us to obtain solutions by using the methods of the fractional calculus [25,26,27,28,82,92,94,104,105]. For some models of economic processes we can derive analytical solutions.…”

An Integro-Differential Equation of the Fractional Form: Cauchy Problem and Solution

“…where the fractional time-derivative D α is taken in the sense of Caputo [8] (see also [6]), and D β is the Riesz operator of order β. In [4] we had already solved Equation (6) for the initial conditions (2).…”